Calculus Examples

$f (x) = \sqrt[5]{x - 5}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x - 5}$ as ${(x - 5)}^{\frac{1}{5}}$ .

$\frac{d}{d x} [{(x - 5)}^{\frac{1}{5}}]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{5}}$ and $g (x) = x - 5$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x - 5$ .

$\frac{d}{d u} [u^{\frac{1}{5}}] \frac{d}{d x} [x - 5]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{5}$ .

$\frac{1}{5} u^{\frac{1}{5} - 1} \frac{d}{d x} [x - 5]$

Step 1.1.2.3

Replace all occurrences of $u$ with $x - 5$ .

$\frac{1}{5} {(x - 5)}^{\frac{1}{5} - 1} \frac{d}{d x} [x - 5]$

Step 1.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1}{5} {(x - 5)}^{\frac{1}{5} - 1 \cdot \frac{5}{5}} \frac{d}{d x} [x - 5]$

Step 1.1.4

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{1}{5} {(x - 5)}^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}} \frac{d}{d x} [x - 5]$

Step 1.1.5

Combine the numerators over the common denominator.

$\frac{1}{5} {(x - 5)}^{\frac{1 - 1 \cdot 5}{5}} \frac{d}{d x} [x - 5]$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $5$ .

$\frac{1}{5} {(x - 5)}^{\frac{1 - 5}{5}} \frac{d}{d x} [x - 5]$

Step 1.1.6.2

Subtract $5$ from $1$ .

$\frac{1}{5} {(x - 5)}^{\frac{- 4}{5}} \frac{d}{d x} [x - 5]$

Step 1.1.7

Combine fractions.

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Step 1.1.7.1

Move the negative in front of the fraction.

$\frac{1}{5} {(x - 5)}^{- \frac{4}{5}} \frac{d}{d x} [x - 5]$

Step 1.1.7.2

Combine $\frac{1}{5}$ and ${(x - 5)}^{- \frac{4}{5}}$ .

$\frac{{(x - 5)}^{- \frac{4}{5}}}{5} \frac{d}{d x} [x - 5]$

Step 1.1.7.3

Move ${(x - 5)}^{- \frac{4}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{5 {(x - 5)}^{\frac{4}{5}}} \frac{d}{d x} [x - 5]$

Step 1.1.8

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$\frac{1}{5 {(x - 5)}^{\frac{4}{5}}} (\frac{d}{d x} [x] + \frac{d}{d x} [- 5])$

Step 1.1.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{5 {(x - 5)}^{\frac{4}{5}}} (1 + \frac{d}{d x} [- 5])$

Step 1.1.10

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$\frac{1}{5 {(x - 5)}^{\frac{4}{5}}} (1 + 0)$

Step 1.1.11

Simplify the expression.

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Step 1.1.11.1

Add $1$ and $0$ .

$\frac{1}{5 {(x - 5)}^{\frac{4}{5}}} \cdot 1$

Step 1.1.11.2

Multiply $\frac{1}{5 {(x - 5)}^{\frac{4}{5}}}$ by $1$ .

$f' (x) = \frac{1}{5 {(x - 5)}^{\frac{4}{5}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{1}{5 {(x - 5)}^{\frac{4}{5}}}$ .

$\frac{1}{5 {(x - 5)}^{\frac{4}{5}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{1}{5 {(x - 5)}^{\frac{4}{5}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{1}{5 {(x - 5)}^{\frac{4}{5}}} = 0$

Step 2.2

Set the numerator equal to zero.

$1 = 0$

Step 2.3

Since $1 \neq 0$ , there are no solutions.

No solution

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{1}{5 \sqrt[5]{{(x - 5)}^{4}}}$

Step 3.2

Set the denominator in $\frac{1}{5 \sqrt[5]{{(x - 5)}^{4}}}$ equal to $0$ to find where the expression is undefined.

$5 \sqrt[5]{{(x - 5)}^{4}} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${(5 \sqrt[5]{{(x - 5)}^{4}})}^{5} = 0^{5}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{{(x - 5)}^{4}}$ as ${(x - 5)}^{\frac{4}{5}}$ .

${(5 {(x - 5)}^{\frac{4}{5}})}^{5} = 0^{5}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${(5 {(x - 5)}^{\frac{4}{5}})}^{5}$ .

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Step 3.3.2.2.1.1

Apply the product rule to $5 {(x - 5)}^{\frac{4}{5}}$ .

$5^{5} {({(x - 5)}^{\frac{4}{5}})}^{5} = 0^{5}$

Step 3.3.2.2.1.2

Raise $5$ to the power of $5$ .

$3125 {({(x - 5)}^{\frac{4}{5}})}^{5} = 0^{5}$

Step 3.3.2.2.1.3

Multiply the exponents in ${({(x - 5)}^{\frac{4}{5}})}^{5}$ .

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Step 3.3.2.2.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3125 {(x - 5)}^{\frac{4}{5} \cdot 5} = 0^{5}$

Step 3.3.2.2.1.3.2

Cancel the common factor of $5$ .

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Step 3.3.2.2.1.3.2.1

Cancel the common factor.

$3125 {(x - 5)}^{\frac{4}{5} \cdot 5} = 0^{5}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$3125 {(x - 5)}^{4} = 0^{5}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Raising $0$ to any positive power yields $0$ .

$3125 {(x - 5)}^{4} = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Divide each term in $3125 {(x - 5)}^{4} = 0$ by $3125$ and simplify.

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Step 3.3.3.1.1

Divide each term in $3125 {(x - 5)}^{4} = 0$ by $3125$ .

$\frac{3125 {(x - 5)}^{4}}{3125} = \frac{0}{3125}$

Step 3.3.3.1.2

Simplify the left side.

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Step 3.3.3.1.2.1

Cancel the common factor of $3125$ .

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Step 3.3.3.1.2.1.1

Cancel the common factor.

$\frac{3125 {(x - 5)}^{4}}{3125} = \frac{0}{3125}$

Step 3.3.3.1.2.1.2

Divide ${(x - 5)}^{4}$ by $1$ .

${(x - 5)}^{4} = \frac{0}{3125}$

Step 3.3.3.1.3

Simplify the right side.

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Step 3.3.3.1.3.1

Divide $0$ by $3125$ .

${(x - 5)}^{4} = 0$

Step 3.3.3.2

Set the $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 3.3.3.3

Add $5$ to both sides of the equation.

$x = 5$

Step 4

Evaluate $\sqrt[5]{x - 5}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 5$ .

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Step 4.1.1

Substitute $5$ for $x$ .

$\sqrt[5]{(5) - 5}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Subtract $5$ from $5$ .

$\sqrt[5]{0}$

Step 4.1.2.2

Rewrite $0$ as $0^{5}$ .

$\sqrt[5]{0^{5}}$

Step 4.1.2.3

Pull terms out from under the radical, assuming real numbers.

$0$

Step 4.2

List all of the points.

$(5, 0)$

Step 5